Integrand size = 20, antiderivative size = 214 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{f}-\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f} \]
-(a+b*arccoth(d*x+c))^2*ln(2/(d*x+c+1))/f+(a+b*arccoth(d*x+c))^2*ln(2*d*(f *x+e)/(-c*f+d*e+f)/(d*x+c+1))/f+b*(a+b*arccoth(d*x+c))*polylog(2,1-2/(d*x+ c+1))/f-b*(a+b*arccoth(d*x+c))*polylog(2,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c +1))/f+1/2*b^2*polylog(3,1-2/(d*x+c+1))/f-1/2*b^2*polylog(3,1-2*d*(f*x+e)/ (-c*f+d*e+f)/(d*x+c+1))/f
Result contains complex when optimal does not.
Time = 13.43 (sec) , antiderivative size = 1767, normalized size of antiderivative = 8.26 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx =\text {Too large to display} \]
(a^2*Log[e + f*x])/f + 2*a*b*(((ArcCoth[c + d*x] - ArcTanh[c + d*x])*Log[e + f*x])/f - (I*(I*ArcTanh[c + d*x]*(-Log[1/Sqrt[1 - (c + d*x)^2]] + Log[I *Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]]) + ((-I)*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])^2 - (I/4)*(Pi - (2*I)*ArcTanh[c + d*x])^2 + 2*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])*Log[1 - E^((2*I)*(I*A rcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x]))] + (Pi - (2*I)*ArcTanh[c + d* x])*Log[1 - E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))] - (Pi - (2*I)*ArcTanh[c + d*x])*Log[2*Sin[(Pi - (2*I)*ArcTanh[c + d*x])/2]] - 2*(I*ArcTanh[(d*e - c *f)/f] + I*ArcTanh[c + d*x])*Log[(2*I)*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTa nh[c + d*x]]] - I*PolyLog[2, E^((2*I)*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTan h[c + d*x]))] - I*PolyLog[2, E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))])/2))/f) - (b^2*(d*e - c*f + f*(c + d*x))*(1 - (c + d*x)^2)*(-1/24*(I*f*Pi^3 - 8*d* e*ArcCoth[c + d*x]^3 - 8*f*ArcCoth[c + d*x]^3 + 8*c*f*ArcCoth[c + d*x]^3 + 24*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] + 24*f*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] - 12*f*PolyLog[3, E^(2*ArcCoth[c + d*x])])/f^2 + ((-(d*e) - f + c*f)*(-(d*e) + f + c*f)*(2*d*e*ArcCoth[c + d*x]^3 - 6*f*ArcCoth[c + d*x]^3 - 2*c*f*ArcCoth[c + d*x]^3 - (4*d*e*Sqrt[ (d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/ E^ArcTanh[f/(d*e - c*f)] + (4*c*f*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f ^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (6*I)...
Time = 0.42 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6662, 27, 6475}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx\) |
\(\Big \downarrow \) 6662 |
\(\displaystyle \frac {\int \frac {d \left (a+b \coth ^{-1}(c+d x)\right )^2}{d \left (e-\frac {c f}{d}\right )+f (c+d x)}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{f (c+d x)-c f+d e}d(c+d x)\) |
\(\Big \downarrow \) 6475 |
\(\displaystyle -\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2 (d e-c f+f (c+d x))}{(d e-c f+f) (c+d x+1)}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 (f (c+d x)-c f+d e)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 (d e-c f+f (c+d x))}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c+d x+1}\right )}{2 f}\) |
-(((a + b*ArcCoth[c + d*x])^2*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcCoth[c + d*x])^2*Log[(2*(d*e - c*f + f*(c + d*x)))/((d*e + f - c*f)*(1 + c + d*x ))])/f + (b*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 + c + d*x)])/f - (b*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - (2*(d*e - c*f + f*(c + d*x)))/( (d*e + f - c*f)*(1 + c + d*x))])/f + (b^2*PolyLog[3, 1 - 2/(1 + c + d*x)]) /(2*f) - (b^2*PolyLog[3, 1 - (2*(d*e - c*f + f*(c + d*x)))/((d*e + f - c*f )*(1 + c + d*x))])/(2*f)
3.2.12.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^2)*(Log[2/(1 + c*x)]/e), x] + (Simp[(a + b*Arc Coth[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b*(a + b*ArcCoth[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcC oth[c*x])*(PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + S imp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyLog[3, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e} , x] && NeQ[c^2*d^2 - e^2, 0]
Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG tQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 9.72 (sec) , antiderivative size = 1603, normalized size of antiderivative = 7.49
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1603\) |
default | \(\text {Expression too large to display}\) | \(1603\) |
parts | \(\text {Expression too large to display}\) | \(1684\) |
1/d*(a^2*d*ln(c*f-d*e-f*(d*x+c))/f-b^2*d*(-ln(c*f-d*e-f*(d*x+c))/f*arccoth (d*x+c)^2-2/f*(-1/2*arccoth(d*x+c)^2*ln(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d* x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)+1/4*I*Pi*csgn(I*(f*c*((d *x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)* f)/((d*x+c+1)/(d*x+c-1)-1))*(csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c +1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I/((d*x+c+1)/(d*x+c-1 )-1))-csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d *x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/((d*x+c+1)/(d*x+c- 1)-1))-csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-( d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1) /(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))+csgn( I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d* x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^2)*arccoth(d*x+c)^2+1/2*arccoth(d*x+ c)^2*ln((d*x+c+1)/(d*x+c-1)-1)-1/2*arccoth(d*x+c)^2*ln(1-1/((d*x+c-1)/(d*x +c+1))^(1/2))-arccoth(d*x+c)*polylog(2,1/((d*x+c-1)/(d*x+c+1))^(1/2))+poly log(3,1/((d*x+c-1)/(d*x+c+1))^(1/2))-1/2*arccoth(d*x+c)^2*ln(1+1/((d*x+c-1 )/(d*x+c+1))^(1/2))-arccoth(d*x+c)*polylog(2,-1/((d*x+c-1)/(d*x+c+1))^(1/2 ))+polylog(3,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+1/2*f*c/(c*f-d*e-f)*arccoth(d *x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/2*f*c/(c*f-d*e -f)*arccoth(d*x+c)*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1...
\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]
\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \]
\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]
a^2*log(f*x + e)/f + integrate(1/4*b^2*(log(1/(d*x + c) + 1) - log(-1/(d*x + c) + 1))^2/(f*x + e) + a*b*(log(1/(d*x + c) + 1) - log(-1/(d*x + c) + 1 ))/(f*x + e), x)
\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2}{e+f\,x} \,d x \]